In this paper, mutual synchronization of two limit-cycle oscillators coupled with delay is studied. The results of bifurcation analysis are presented under the assumption that the delay is small in comparison with the oscillation build-up time. The stability conditions for in-phase and anti-phase modes of synchronization are analyzed at different values of parameters. The synchronization tongues on the frequency mismatch — coupling strength plane are presented.

In this paper, the Selkov model describing glycolytic oscillations of substrate and product is considered.We have obtained a parametric zone of equilibria where, depending on the initial data, two types of transients are observed. It is shown that in this zone the system is highly sensitive even to small random perturbations. We demonstrate and study the phenomenon of stochastic generation of large-amplitude oscillations in the equilibrium zone. In the study of probability distributions of random trajectories of the forced system, it is shown that this phenomenon is associated with a stochastic $P$-bifurcation. The deformations of the frequency characteristics are confirmed by spectral analysis. It is shown that in the regime of the stochastic excitation of stable equilibrium, the dominant frequency of the noise-induced spikes practically coincides with the frequency of the deterministic relaxation oscillations observed just after the Andronov–Hopf bifurcation.

An autonomous dynamical system with one degree of freedom with a cylindrical phase space is studied. The mathematical model of the system is given by a second-order differential equation that contains terms responsible for nonconservative forces. A coefficient $\alpha$ at these terms is supposed to be a small parameter of the model. So the system is close to a Hamiltonian one.
In the first part of the paper, it is additionally supposed that one of nonconservative terms corresponds to dissipative or to antidissipative forces, and coefficient $b$ at this term is a varied parameter. The Poincaré – Pontryagin approach is used to construct a bifurcation diagram of periodic trajectories with respect to the parameter b for sufficiently small values of $\alpha$.
In the second part of the paper, a system with nonconservative terms of general form is studied. Two supplementary systems of special form are constructed. Results of the first part of the paper are applied to these systems. Comparison of bifurcation diagrams for these supplementary systems has allowed deriving necessary conditions for the existence of periodic trajectories in the initial system for sufficiently small $\alpha$.
The third part of the paper contains an example of the study of periodic trajectories of one system, which, for zero value of the small parameter, coincides with a Hamiltonian system $H_0$. It is proved that there exist periodic trajectories which do not satisfy the Poincaré – Pontryagin sufficient conditions for emergence of periodic trajectories from trajectories of the system $H_0$.

A mathematical model of contact interaction of two geometrically nonlinear S.P. Timoshenko beams is obtained. The transverse alternating load acts on one of the beams. The infinitedimensional problem is reduced to a finite-dimensional one by using the second-order finite difference method. The Cauchy problem obtained is solved by the Runge – Kutta method of 4th order. The contact pressure is determined by B.Ya. Kantor’s method. The analysis of the results is carried out by methods of nonlinear dynamics and qualitative theory of differential equations. The scenario of transition of vibrations of the structure from harmonic to chaotic ones is studied. It is found that the shape of the beams’ vibrations becomes asymmetric in the event of the first bifurcation, but at the same time, there comes a phase synchronization of chaotic vibrations. Maps of the dynamic behavior for both beams are constructed.

Equilibrium states of Arakawa approximations of a two-dimensional incompressible inviscid fluid are investigated in the case of high resolution $8192^2$. Comparison of these states with quasiequilibrium states of a viscid fluid is made. Special attention is paid to the stepped shape of large coherent structures and to the presence of small vortices in final states. It is shown that the large-scale dynamics of Arakawa approximations are similar to the theoretical predictions for an ideal fluid. Cesaro convergence is investigated as an alternative technique to get condensed states. Additionally, it can be used to solve the problem of nonstationary final states.

We consider the motion of Lagrange’s top with a suspension point performing the specified highfrequency periodic motion with small amplitude in three-dimensional space. The approximate autonomous system of equations of motion written in the form of canonical Hamiltonian equations is investigated. The problem of the existence and number of stationary rotations of the top about its dynamical symmetry axis is solved. The study of stability of the corresponding equilibrium positions of the reduced two-degree-of-freedom system for fixed values of the cyclic integral constant depending on the angular velocity of rotation is carried out. For suspension points’ motions allowing for stationary rotations about the vertical, a detailed linear and nonlinear stability analysis of these rotations and rotations about inclined axes is carried out. For a number of other cases of the suspension point motions a linear stability analysis is carried out.

An analysis is presented of the nonlinear dynamics of harmonically coupled pendulums without restrictions to oscillation amplitudes. This is a basic model in many areas of mechanics and physics (paraffin crystals, DNA molecules etc.). Stationary solutions of equations of motion corresponding to nonlinear normal modes (NNMs) are obtained. The inversion of the NNM frequencies with increasing oscillation amplitude is found. An essentially nonstationary process of the resonant energy exchange is described in terms of limiting phase trajectories (LPTs), for which an effective analytic representation is obtained in slow time-scale. Explicit expressions of threshold values of dimensionless parameters are found which correspond to the instability of NNMs and to the transition (in parametric space) from the full energy exchange between the pendulums to the localization of energy. The analytic results obtained are verified by analysis of the Poincar´e sections describing evolution of the initial system.

In the framework of the Jacobi method we obtain a new integrable system on the plane with a natural Hamilton function and a second integral of motion which is a polynomial of sixth order in momenta. The corresponding variables of separation are images of usual parabolic coordinates on the plane after a suitable Bäcklund transformation. We also present separated relations and prove that the corresponding vector field is bi-Hamiltonian.

This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.